... inequalities. The achieved results is inspired by the paper of Hara et al. [J.Inequal. Appl. 2, 387-395, (1998)], and the methods from Guan [Math. Inequal. Appl.9, 797-805, (2006)]. The inequalities ... r is a positive integer. In [2], Guan proved that both Fn(x,r) andF∗n(x, r)are Schur concave in Rn+. The mainof this paper is to investigate the Schur convexity for the functionsFn(x, ... analytic inequalities by use of the theory ofmajorization. For convenience of readers, we recall some definitions as follows, which can befound in many references, such as [3].Qian Journal of Inequalities...